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Symmetric structures on a closed curve. (English) Zbl 0778.30045
It is shown that the quasisymmetric topology of Ahlfors (the topology coming from uniform ratio distortion) on local homeomorphisms in one real dimension is defined when, and only when, the underlying one-manifold is provided with a “symmetric structure”, one defined by using a structure pseudogroup the quasisymmetric closure of the $$C^ 1$$-diffeomorphisms of the real line. It is shown that the set of all symmetric structures on a closed curve compatible with a background quasisymmetric structure is naturally a complete, complex Banach manifold, modelled on the Banach space $$\Lambda^*/\lambda^*$$, where $$\Lambda^*$$ and $$\lambda^*$$ are the spaces of continuous functions $$F$$ on the circle introduced by Zygmund in 1945; $\Lambda^*:\;F(x+t)+F(x-t)-2F(x)=O(t), \qquad \lambda^*:\;F(x+t)+F(x-t)-2F(x)=o(t)$ and the complex structure is given by the Hilbert transform. The discussion covers analytical and geometrical properties of symmetric homeomorphisms and symmetric quasicircles and suggests how the Bers’ embedding technique (1965) may be used in a variety of contexts.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
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